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Physical Review

Exact Inversion of the Fugacity-Density Relation for Ideal Quantum Gases

A. Léonard

Published Nov 5, 1968• 16 citations
Publisher

Abstract

The fugacities of ideal Fermi and Bose gases are calculated explicity as a function of density and temperature, using the analytic properties of the integral appearing in the fugacity-density equation. The Hilbert problem of the theory of analytic functions is encountered.
DOI
10.1103/physrev.175.221

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Calculations of chemical potentials for ideal monatomic gases with Bose-Einstein and Fermi-Dirac statistics as functions of temperature, across the temperature region that is typical for the collective quantum degeneracy effect, are presented. Numerical calculations are performed without any additional approximations, and explicit dependences of the chemical potentials on temperature are constructed at a fixed density of gas particles. Approximate polynomial dependences of chemical potentials on temperature are obtained that allow for the results to be used in further studies without re-applying the involved numerical methods. The ease of using the obtained representations is demonstrated on examples of deformation of distribution for a population of energy states at low temperatures, and on the impact of quantum statistics (exchange interaction) on the equations of state for ideal gases and some of the thermodynamic properties thereof. The results of this study essentially unify two opposite limiting cases in an intermediate region that are used to describe the equilibrium states of ideal gases, which are well known from university courses on statistical physics, thus adding value from an educational point of view.
For an ideal statistical gas, where PV=sEPV=sEPV=sE, all thermodynamic functions can be derived from G(V,T)G(V,T)G(V,T), i.e., A(V,T)A(V,T)A(V,T) is given by a suitable integration over G(V,T)G(V,T)G(V,T). Since $G={N}_{\ensuremath{\mu}}$, the normalization constant, $\ensuremath{\mu}(V,T)$, of the distribution function is sufficient to provide a complete thermodynamic description of such a system. With the aid of the $\ensuremath{\aleph}$-theorem it is shown that there is a sharp physical distinction between the degenerate and non-degenerate Bose-Einstein gases; to any given density there corresponds a transition temperature between them. The method developed is applied to the general (relativistic and non-relativistic) non-degenerate quantum and degenerate Fermi-Dirac gases.